Computational Cognitive Neuroscience

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Computational Cognitive Neuroscience PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Wed, 22 Jan 2014 05:51:52 CETContents Articles 0. Frontmatter 1 CCNBook/Frontmatter 1 CCNBook/Contributors 2 1. Introduction 3 CCNBook/Intro 3 Part I Basic Computational Mechanisms 9 2. The Neuron 10 CCNBook/Neuron 10 CCNBook/Neuron/Biology 25 CCNBook/Neuron/Electrophysiology 27 3. Networks 30 CCNBook/Networks 30 4. Learning Mechanisms 48 CCNBook/Learning 48 Part II Cognitive Neuroscience 67 5. Brain Areas 68 CCNBook/BrainAreas 68 6. Perception and Attention 78 CCNBook/Perception 78 7. Motor Control and Reinforcement Learning 95 CCNBook/Motor 95 8. Learning and Memory 109 CCNBook/Memory 109 9. Language 123 CCNBook/Language 12310. Executive Function 138 CCNBook/Executive 138 References Article Sources and Contributors 160 Image Sources, Licenses and Contributors 1611 0. Frontmatter CCNBook/Frontmatter Citation Here is the citation for this book, in standard APA format: O'Reilly, R. C., Munakata, Y., Frank, M. J., Hazy, T. E., and Contributors (2012). Computational Cognitive Neuroscience. Wiki Book, 1st Edition. URL: http:/ / ccnbook. colorado. edu and in BibTeX format: BOOKOReillyMunakataFrankEtAl12, author=Randall C. O'Reilly and Yuko Munakata and Michael J. Frank and Thomas E. Hazy and Contributors, title=Computational Cognitive Neuroscience, year=2012, publisher=Wiki Book, 1st Edition, URL: \urlhttp://ccnbook.colorado.edu, url=http://ccnbook.colorado.edu, Copyright and Licensing The contents of this book are Copyright © O'Reilly and Munakata, 2012. The book contents constitute everything within the CCNBook/ prefix on this wiki, and all figures that are linked on these pages. The following license is in effect for use of the text outside of this wiki: html a rel="license" href="http://creativecommons.org/licenses/by-sa/3.0/"img alt="Creative Commons License" style="border-width:0" src="http://i.creativecommons.org/l/by-sa/3.0/88x31.png" //a This work is licensed under a a rel="license" href="http://creativecommons.org/licenses/by-sa/3.0/"Creative Commons Attribution-ShareAlike 3.0 Unported License/a. /htmlCCNBook/Contributors 2 CCNBook/Contributors Back to Main Page Please contribute to this book One of the great advantages of the wiki format is that many people can contribute to produce something that should hopefully benefit from lots of different perspectives and ideas. O'Reilly & Munakata will retain final editorial control, by reviewing edits that are made, etc. If you would like to contribute, smaller changes and additions can be made directly into the text, and larger contributions should be discussed on the associated Discussion page for the relevant Chapter(s). Each person is responsible for updating their own summary of contributions on this page (and contributions are automatically tracked by the wiki as well). For pragmatic purposes, you must yield the copyright to your contributions, with the understanding that everything is being made publicly available through a Creative Commons license as shown below. Please update your contributions to the text here Major Contributors (Authors) • Randall C. O'Reilly: primary initial author of text and simulation exercises, and author of prior CECN book upon which this is based. • Yuko Munakata: planning of text and editing of rough drafts, and author of prior CECN book upon which this is based. • Tom Hazy: wrote first draft of Executive Function Chapter and did major work converting simulation docs to new format. • Michael J. Frank: edits/additions to text in various chapters and simulations, and provided the basal ganglia simulation for the Motor and Reinforcement Learning Chapter. Note: Major contributors are considered co-authors, as reflected in the correct Citation for this book. Additional major contributors will be added to a subsequent edition of the book. Other Contributors • Trent Kriete: updating of sims docs. • Sergio Verduzco: I-F curves in the Neuron Chapter. • • Several students from O'Reilly's class in Spring 2011, who suffered through the initial writing process.3 1. Introduction CCNBook/Intro You are about to embark on one of the most fascinating scientific journeys possible: inside your own brain We start this journey by understanding what individual neurons in your neocortex do with the roughly 10,000 synaptic input signals that they receive from other neurons. The neocortex is the most evolutionarily recent part of the brain, which is also most enlarged in humans, and is where most of your thinking takes place. The numbers of neurons and synapses between neurons in the neocortex are astounding: roughly 20 billion neurons, each of which is interconnected with roughly 10,000 others. That is several times more neurons than people on earth. And each neuron is far more social than we are as people estimates of the size of stable human social networks are only around 150-200 people, compared to the 10,000 for neurons. We've got a lot going on under the hood. At these scales, the influence of any one neuron on any other is relatively small. We'll see that these small influences can be shaped in powerful ways through learning mechanisms, to achieve complex and powerful forms of information processing. And this information processing prowess does not require much complexity from the individual neurons themselves fairly simple forms of information integration both accurately describe the response properties of actual neocortical neurons, and enable sophisticated information processing at the level of aggregate neural networks. After developing an understanding of these basic neural information processing mechanisms in Part I of this book, we continue our journey in Part II by exploring many different aspects of human thought (cognition), including perception and attention, motor control and reinforcement learning, learning and memory, language, and executive function. Amazingly, all these seemingly different cognitive functions can be understood using a small set of common neural mechanisms. In effect, our neocortex is a fantastic form of silly putty, which can be molded by the learning process to take on many different cognitive tasks. For example, we will find striking similarities across different brain areas and cognitive functions the development of primary visual cortex turns out to tell us a lot about the development of rich semantic knowledge of word meanings Some Phenomena We'll Explore Here is a list of some of the cognitive neuroscience phenomena we'll explore in Part II of the book: • Vision: We can effortlessly recognize countless people, places, and things. Why is this so hard for robots? We will explore this issue in a network that views natural scenes (mountains, trees, etc.), and develops brain-like ways of encoding them using principles of learning. • Attention: Where's Waldo? We'll see in a model how two visual processing pathways work together to help focus our attention in different locations in space (whether we are searching for something or just taking things in), and why damage to one of these pathways leads people to ignore half of space. • Dopamine and Reward: Why do we get bored with things so quickly? Because our dopamine system is constantly adapting to everything we know, and only gives us rewards when something new or different occurs. We'll see how this all happens through interacting brain systems that drive phasic dopamine release. • Episodic memory: How can damage to a small part of our brain cause amnesia? We'll see how in a model that replicates the structure of the hippocampus. This model provides insight into why the rest of the brain isn't well-suited to take on the job of forming new episodic memories. • Reading: What causes dyslexia, and why do people who have it vary so much in their struggles with reading? We'll explore these issues in a network that learns to read and pronounce nearly 3,000 English words, andCCNBook/Intro 4 generalizes to novel nonwords (e.g., “mave” or “nust”) just like people do. We'll see why damaging the network in different ways simulates various forms of dyslexia. • Meaning: "A rose is a rose is a rose." 1 But how do we know what a rose is in the first place? We'll explore this through a network that “reads” every paragraph in a textbook, and acquires a surprisingly good semantic understanding by noting which words tend to be used together or in similar contexts. • Task directed behavior: How do we stay focused on tasks that we need to get done or things that we need to pay attention to, in the face of an ever-growing number of distractions (like email, text messages, and tweets)? We'll explore this issue through a network that simulates the “executive” part of the brain, the prefrontal cortex. We will see how this area is uniquely-suited to protect us from distraction, and how this can change with age. The Computational Approach An important feature of our journey through the brain is that we use the vehicle of computer models to understand cognitive neuroscience (i.e., Computational Cognitive Neuroscience). These computer models enrich the learning experience in important ways we routinely hear from our students that they didn't really understand anything until they pulled up the computer model and played around with it for a few hours. Being able to manipulate and visualize the brain using a powerful 3D graphical interface brings abstract concepts to life, and enables many experiments to be conducted easily, cleanly, and safely in the comfort of your own laptop. This stuff is fun, like a video game think "sim brain", as in the popular "sim city" game from a few years ago. At a more serious level, the use of computer models to understand how the brain works has been a critical contributor to scientific progress in this area over the past few decades. A key advantage of computer modeling is its ability to wrestle with complexity that often proves daunting to otherwise unaided human understanding. How could we possibly hope to understand how billions of neurons interacting with 10's of thousands of other neurons produce complex human cognition, just by talking in vague verbal terms, or simple paper diagrams? Certainly, nobody questions the need to use computer models in climate modeling, to make accurate predictions and understand how the many complex factors interact with each other. The situation is only more dire in cognitive neuroscience. Nevertheless, in all fields where computer models are used, there is a fundamental distrust of the models. They are themselves complex, created by people, and have no necessary relationship to the real system in question. How do we know these models aren't just completely made up fantasies? The answer seems simple: the models must be constrained by data at as many levels as possible, and they must generate predictions that can then be tested empirically. In what follows, we discuss different approaches that people might take to this challenge this is intended to give a sense of the scientific approach behind the work described in this book as a student this is perhaps not so relevant, but it might help give some perspective on how science really works. In an ideal world, one might imagine that the neurons in the neural model would be mirror images of those in the actual brain, replicating as much detail as is possible given the technical limitations for obtaining the necessary details. They would be connected exactly as they are in the real brain. And they would produce detailed behaviors that replicate exactly how the organism in question behaves across a wide range of different situations. Then you would feel confident that your model is sufficiently "real" to trust some of its predictions. But even if this were technically feasible, you might wonder whether the resulting system would be any more comprehensible than the brain itself In other words, we would only have succeeded in transporting the fundamental mysteries from the brain into our model, without developing any actual understanding about how the thing really works. From this perspective, the most important thing is to develop the simplest possible model that captures the most possible data this is basically the principle of Ockham's razor, which is widely regarded as a central principle for all scientific theorizing. In some cases, it is easy to apply this razor to cut away unnecessary detail. Certainly many biological properties of neurons are irrelevant for their core information processing function (e.g., cellular processes that are common to allCCNBook/Intro 5 biological cells, not just neurons). But often it comes down to a judgment call about what phenomena you regard as being important, which will vary depending on the scientific questions being addressed with the model. The approach taken for the models in this book is to find some kind of happy (or unhappy) middle ground between biological detail and cognitive functionality. This middle ground is unhappy to the extent that researchers concerned with either end of this continuum are dissatisfied with the level of the models. Biologists will worry that our neurons and networks are overly simplified. Cognitive psychologists will be concerned that our models are too biologically detailed, and they can make much simpler models that capture the same cognitive phenomena. We who relish this "golden middle" ground are happy when we've achieved important simplifications on the neural side, while still capturing important cognitive phenomena. This level of modeling explores how consideration of neural mechanisms inform the workings of the mind, and reciprocally, how cognitive and computational constraints afford a richer understanding of the problems these mechanisms evolved to solve. It can thus make predictions for how a cognitive phenomenon (e.g., memory interference) is affected by changes at the neural level (due to disease, pharmacology, genetics, or similarly due to changes in the cognitive task parameters). The model can then be tested, falsified and refined. In this sense, a model of cognitive neuroscience is just like any other 'theory', except that it is explicitly specified and formalized, forcing the modeler to be accountable for their theory if/when the data don't match up. Conversely, models can sometimes show that when an existing theory is faced with challenging data, the theory may hold up after all due to a particular dynamic that may not be considered from verbal theorizing. Ultimately, it comes down to aesthetic or personality-driven factors, which cause different people to prefer different overall strategies to computer modeling. Each of these different approaches has value, and science would not progress without them, so it is fortunate that people vary in their personalties so different people end up doing different things. Some people value simplicity, elegance, and cleanliness most highly these people will tend to favor abstract mathematical (e.g., Bayesian) cognitive models. Other people value biological detail above all else, and don't feel very comfortable straying beyond the most firmly established facts they will prefer to make highly elaborated individual neuron models incorporating everything that is known. To live in the middle, you need to be willing to take some risks, and value most highly the process of emergence, where complex phenomena can be shown to emerge from simpler underlying mechanisms. The criteria for success here are a bit murkier and subjective basically it boils down to whether the model is sufficiently simple to be comprehensible, but not so simple as to make its behavior trivial or otherwise so fully transparent that it doesn't seem to be doing you any good in the first place. One last note on this issue is that the different levels of models are not mutually exclusive. Each of the low level biophysical and high level cognitive models have made enormous contributions to understanding and analysis in their respective domains (much of which is a basis for further simplification or elaboration in the book). In fact, much ground can be (and to some extent already has been) gained by attempts to understand one level of modeling in terms of the other. At the end of the day, linking from molecule to mind spans multiple levels of analysis, and like studying the laws of particle physics to planetary motion, require multiple formal tools.CCNBook/Intro 6 Emergent Phenomena What makes something a satisfying scientific explanation? A satisfying answer is that you can explain a seemingly complex phenomenon in terms of simpler underlying mechanisms, that interact in specific ways. The classic scientific process of reductionism plays a critical role here, where the Figure 1.1: Simple example of emergence of phenomena, in a very complex system is reduced to simpler parts. However, simple physical system: two gears. Both panel a and b contain the one also needs to go in the opposite, oft-neglected same parts. Only panel b exhibits an emergent phenomena through direction, reconstructionism, where the complex the interaction of the two gears, causing things like torque and speed system is actually reconstructed from these simpler differentials on the two different gears. Emergence is about interactions between parts. Computer models can capture many parts. Often the only way to practically achieve this complex interactions and reveal nonobvious kinds of emergence. reconstruction is through computational modeling. The result is an attempt to capture the essence of emergence. Emergence can be illustrated in a very simple physical system, two interacting gears, as shown in Figure 1.1. It is not mysterious or magical. On the other hand, it really is. You can make the gears out of any kind of sufficiently hard material, and they will still work. There might be subtle factors like friction and durability that vary. But over a wide range, it doesn't matter what the gears are made from. Thus, there is a level of transcendence that occurs with emergence, where the behavior of the more complex interacting system does not depend on many of the detailed properties of the lower level parts. In effect, the interaction itself is what matters, and the parts are mere place holders. Of course, they have to be there, and meet some basic criteria, but they are nevertheless replaceable. Taking this example into the domain of interest here, does this mean that we can switch out our biological neurons for artificial ones, and everything should still function the same, as long as we capture the essential interactions in the right way? Some of us believe this to be the case, and that when we finally manage to put enough neurons in the right configuration into a big computer simulation, the resulting brain will support consciousness and everything else, just like the ones in our own heads. One interesting further question arises: how important are all the interactions between our physical bodies and the physical environment? There is good reason to believe that this is critical. Thus, we'll have to put this brain in a robot. Or perhaps more challengingly, in a virtual environment in a virtual reality, still stuck inside the computer. It will be fascinating to ponder this question on your journey through the simulated brain... Why Should We Care about the Brain? One of the things you'll discover on this journey is that Computational Cognitive Neuroscience is hard. There is a lot of material at multiple levels to master. We get into details of ion channels in neurons, names of pathways in different parts of the brain, effects of lesions to different brain areas, and patterns of neural activity, on top of all the details about behavioral paradigms and reaction time patterns. Wouldn't it just be a lot simpler if we could ignore all these brain details, and just focus on what we really care about how does cognition itself work? By way of analogy, we don't need to know much of anything about how computer hardware works to program in Visual Basic or Python, for example. Vastly different kinds of hardware can all run the same programming languages and software. Can't we just focus on the software of the mind and ignore the hardware? Exactly this argument has been promulgated in many different forms over the years, and indeed has a bit of a resurgence recently in the form of abstract Bayesian models of cognition. David Marr was perhaps the most influential in arguing that one can somewhat independently examine cognition at three different levels:CCNBook/Intro 7 • Computational what computations are being performed? What information is being processed? • Algorithmic how are these computations being performed, in terms of a sequence of information processing steps? • Implementational how does the hardware actually implement these algorithms? This way of dividing up the problem has been used to argue that one can safely ignore the implementation (i.e., the brain), and focus on the computational and algorithmic levels, because, like in a computer, the hardware really doesn't matter so much. However, the key oversight of this approach is that the reason hardware doesn't matter in standard computers is that they are all specifically designed to be functionally equivalent in the first place Sure, there are lots of different details, but they are all implementing a basic serial Von Neumann architecture. What if the brain has a vastly different architecture, which makes some algorithms and computations work extremely efficiently, while it cannot even support others? Then the implementational level would matter a great deal. There is every reason to believe that this is the case. The brain is not at all like a general purpose computational device. Instead, it is really a custom piece of hardware that implements a very specific set of computations in massive parallelism across its 20 billion neurons. In this respect, it is much more like the specialized graphics processing units (GPU's) in modern computers, which are custom designed to efficiently carry out in massive parallelism the specific computations necessary to render complex 3D graphics. More generally, the field of computer science is discovering that parallel computation is exceptionally difficult to program, and one has to completely rethink the algorithms and computations to obtain efficient parallel computation. Thus, the hardware of the brain matters a huge amount, and provides many important clues as to what kind of algorithms and computations are being performed. Historically, the "ignore the brain" approaches have taken an interesting trajectory. In the 1960's through the early 1990's, the dominant approach was to assume that the brain actually operates much like a standard computer, and researchers tended to use concepts like logic and symbolic propositions in their cognitive models. Since then, a more statistical metaphor has become popular, with the Bayesian probabilistic framework being widely used in particular. This is an advance in many respects, as it emphasizes the graded nature of information processing in the brain (e.g., integrating various graded probabilities to arrive at an overall estimate of the likelihood of some event), as contrasted with hard symbols and logic, which didn't seem to be a particularly good fit with the way that most of cognition actually operates. However, the actual mathematics of Bayesian probability computations are not a particularly good fit to how the brain operates at the neural level, and much of this research operates without much consideration for how the brain actually functions. Instead, a version of Marr's computational level is adopted, by assuming that whatever the brain is doing, it must be at least close to optimal, and Bayesian models can often tell us how to optimally combine uncertain pieces of information. Regardless of the validity of this optimality assumption, it is definitely useful to know what the optimal computations are for given problems, so this approach certainly has a lot of value in general. However, optimality is typically conditional on a number of assumptions, and it is often difficult to decide among these different assumptions. If you really want to know for sure how the brain is actually producing cognition, clearly you need to know how the brain actually functions. Yes, this is hard. But it is not impossible, and the state of neuroscience these days is such that there is a wealth of useful information to inform all manner of insights into how the brain actually works. It is like working on a jigsaw puzzle the easiest puzzles are full of distinctive textures and junk everywhere, so you can really see when the pieces fit together. The rich tableau of neuroscience data provides all this distinctive junk to constrain the process of puzzling together cognition. In contrast, abstract, purely cognitive models are like a jigsaw puzzle with only a big featureless blue sky. You only have the logical constraints of the piece shapes, which are all highly similar and difficult to discriminate. It takes forever. A couple of the most satisfying instances of all the pieces coming together to complete a puzzle include:CCNBook/Intro 8 • The detailed biology of the hippocampus, including high levels of inhibition and broad diffuse connectivity, fit together with its unique role in rapidly learning new episodic information, and the remarkable data from patient HM who had his hippocampus resected to prevent intractable epilepsy. Through computational models in the Memory Chapter, we can see that these biological details produce high levels of pattern separation which keep memories highly distinct, and thus enable rapid learning without creating catastrophic levels of interference. • The detailed biology of the connections between dopamine, basal ganglia, and prefrontal cortex fit together with the computational requirements for making decisions based on prior reward history, and learning what information is important to hold on to, versus what can be ignored. Computational models in the Executive Function Chapter show that the dopamine system can exhibit a kind of time travel needed to translate later utility into an earlier decision of what information to maintain, and those in the Motor Chapter show that the effects of dopamine on the basal ganglia circuitry are just right to facilitate decision making based on both positive and negative outcomes. And the interaction between the basal ganglia and the prefrontal cortex enables basal ganglia decisions to influence what is maintained and acted upon in the prefrontal cortex. There are a lot of pieces here, but the fact that they all fit together so well into a functional model and that many aspects of them have withstood the test of direct experimentation makes it that much more likely that this is really what is going on. How to Read this Book This book is intended to accommodate many different levels of background and interests. The main chapters are relatively short, and provide a high-level introduction to the major themes. There will be an increasing number of detailed subsections added over time, to support more advanced treatment of specific issues. The ability to support these multiple levels of readers is a major advantage of the wiki format. We also encourage usage of this material as an adjunct for other courses on related topics. The simulation models can be used by themselves in many different courses. Due to the complexity and interconnected nature of the material (mirroring the brain itself), it may be useful to revisit earlier chapters after having read later chapters. Also, we strongly recommend reading the Brain Areas chapter now, and then re-reading it in its regular sequence after having made it all the way through Part I. It provides a nice high-level summary of functional brain organization, that bridges the two parts of the book, and gives an overall roadmap of the content we'll be covering. Some of it won't make as much sense until after you've read Part I, but doing a quick first read now will provide a lot of useful perspective. External Resources • Gary Cottrell's solicited compilation of important computational modeling papers 2 References 1 http:/ / en. wikipedia. org/ wiki/ Rose_is_a_rose_is_a_rose_is_a_rose 2 http:/ / cseweb. ucsd. edu/ gary/ cse258a/ CogSciLiterature. html9 Part I Basic Computational Mechanisms10 2. The Neuron CCNBook/Neuron One major reason the brain can be so plastic and learn to do so many different things, is that it is made up of a highly-sculptable form of silly putty: billions of individual neurons that are densely interconnected with each other, and capable of shaping what they do by changing these patterns of interconnections. The brain is like a massive LEGO set, where each of the individual pieces is quite simple (like a single LEGO piece), and all the power comes from the nearly infinite ways that these simple pieces can be recombined to do different things. Figure 2.1: Trace of a simulated neuron spiking action potentials in response to an So the good news for you the student excitatory input the yellow v_m membrane potential (voltage of the neuron) increases is, the neuron is fundamentally simple. (driven by the excitatory net input) until it reaches threshold (around .5), at which point a Lots of people will try to tell you green act activation spike (action potential) is triggered, which then resets the membrane otherwise, but as you'll see as you go potential back to its starting value (.3) and the process continues. The spike is communicated other neurons, and the overall rate of spiking (tracked by the blue act_eq through this book, simple neurons can value) is proportional to the level of excitatory net input (relative to other opposing account for much of what we know factors such as inhibition the balance of all these factors is reflected in the net current about how the brain functions. So, I_net). You can produce this graph and manipulate all the relevant parameters in the even though they have a lot of moving Neuron exploration for this chapter. parts and you can spend an entire career learning about even just one tiny part of a neuron, we strongly believe that all this complexity is in the service of a very simple overall function. What is that function? Fundamentally, it is about detection. Neurons receive thousands of different input signals from other neurons, looking for specific patterns that are "meaningful" to them. A very simple analogy is with a smoke detector, which samples the air and looks for telltale traces of smoke. When these exceed a specified threshold limit, the alarm goes off. Similarly, the neuron has a threshold and only sends an "alarm" signal to other neurons when it detects something significant enough to cross this threshold. The alarm is called an action potential or spike and it is the fundamental unit of communication between neurons. Our goal in this chapter is to understand how the neuron receives input signals from other neurons, integrates them into an overall signal strength that is compared against the threshold, and communicates the result to other neurons. We will see how these processes can be characterized mathematically in computer simulations (summarized in Figure 2.1). In the rest of the book, we will see how this simple overall function of the neuron ultimately enables us to perceive the world, to think, to communicate, and to remember.CCNBook/Neuron 11 Math warning: This chapter and the Learning Mechanisms Chapter are the only two in the entire book with significant amounts of math (because these two chapters describe in detail the equations for our simulations). We have separated the conceptual from the mathematical content, and those with an aversion to math can get by without understanding all the details. So, don't be put off or overwhelmed by the math here Basic Biology of a Neuron as Detector Figure 2.2 shows the correspondence between neural biology and the detection functions they serve. Synapses are the connection points between sending neurons (the ones firing an alarm and sending a signal) and receiving neurons (the ones receiving that signal). Most synapses are on dendrites, which are the large branching trees (the word "dendrite" is derived from the Greek "dendros," meaning tree), which is where the Figure 2.2: Neuron as a detector, with corresponding biological components. neuron integrates all the input signals. Like tributaries flowing into a major river, all these signals flow into the main dendritic trunk and into the cell body, where the final integration of the signal takes place. The thresholding takes place at the very start of the output-end of the neuron, called the axon (this starting place is called the axon hillock apparently it looks like a little hill or something). The axon also branches widely and is what forms the other side of the synapses onto other neuron's dendrites, completing the next chain of communication. And onward it goes. This is all you need to know about the neuron biology to understand the basic detector functionality: It just receives inputs, integrates them, and decides whether the integrated input is sufficiently strong to trigger an output signal. There are some additional biological properties regarding the nature of the input signals, which we'll see have various implications for neural function, including making the integration process better able to deal with large changes in overall input signal strength. There are at least three major sources of input signals to the neuron: • Excitatory inputs these are the "normal", most prevalent type of input from other neurons (roughly 85% of all inputs), which have the effect of exciting the receiving neuron (making it more likely to get over threshold and fire an "alarm"). They are conveyed via a synaptic channel called AMPA, which is opened by the neurotransmitter glutamate. • Inhibitory inputs these are the other 15% of inputs, which have the opposite effect to the excitatory inputs they cause the neuron to be less likely to fire, and serve to make the integration process much more robust by keeping the excitation in check. There are specialized neurons in the brain called inhibitory interneurons that generate this inhibitory input (we'll learn a lot more about these in the Networks chapter). This input comes in via GABA synaptic channels, driven by the neurotransmitter GABA. • Leak inputs these aren't technically inputs, as they are always present and active, but they serve a similar function to the inhibitory inputs, by counteracting the excitation and keeping the neuron in balance overall. Biologically, leak channels are potassium channels (K). The inhibitory and excitatory inputs come from different neurons in the cortex: a given neuron can only send either excitatory or inhibitory outputs to other neurons, not both. We will see the multiple implications of this constraint throughout the text.CCNBook/Neuron 12 Finally, we introduce the notion of the net synaptic efficacy or weight, which represents the total impact that a sending neuron activity signal can have on the receiving neuron, via its synaptic connection. The synaptic weight is one of the most important concepts in the entire field of computational cognitive neuroscience We will be exploring it in many different ways as we go along. Biologically, it represents the net ability of the sending neuron's action potential to release neurotransmitter, and the ability of that neurotransmitter to open synaptic channels on the postsynaptic side (including the total number of such channels that are available to be opened). For the excitatory inputs, it is thus the amount of glutamate released by the sending neuron into the synapse, and the number and efficacy of AMPA channels on the receiving neuron's side of the synapse. Computationally, the weights determine what a neuron is detecting. A strong weight value indicates that the neuron is very sensitive to that particular input neuron, while a low weight means that that input is relatively unimportant. The entire process of Learning amounts to changing these synaptic weights as a function of neural activity patterns in the sending and receiving neurons. In short, everything you know, every cherished memory in your brain, is encoded as a pattern of synaptic weights To learn more about the biology of the neuron, see Neuron/Biology. Dynamics of Integration: Excitation vs. Inhibition and Leak The process of integrating the three different types of input signals (excitation, inhibition, leak) lies at the heart of neural computation. This section provides a conceptual, intuitive understanding of this process, and how it relates to the underlying electrical properties of neurons. Later, we'll see how to translate this process into mathematical equations that can actually be simulated on the computer. Figure 2.3: The neuron is a tug-of-war battleground between inhibition and excitation The integration process can be the relative strength of each is what determines the membrane potential, Vm, which is understood in terms of a tug-of-war what must get over threshold to fire an action potential output from the neuron. (Figure 2.3). This tug-of-war takes place in the space of electrical potentials that exist in the neuron relative to the surrounding extracellular medium in which neurons live (interestingly, this medium, and the insides of neurons and other cells as well, is basically salt water with sodium (Na+), chloride (Cl-) and other ions floating around we carry our remote evolutionary environment around within us at all times). The core function of a neuron can be understood entirely in electrical terms: voltages (electrical potentials) and currents (flow of electrically charged ions in and out of the neuron through tiny pores called ion channels). To see how this works, let's just consider excitation versus inhibition (inhibition and leak are effectively the same for our purposes at this time). The key point is that the integration process reflects the relative strength of excitation versus inhibition if excitation is stronger than inhibition, then the neuron's electrical potential (voltage) increases, perhaps to the point of getting over threshold and firing an output action potential. If inhibition is stronger, then the neuron's electrical potential decreases, and thus moves further away from getting over the threshold for firing. Before we consider specific cases, let's introduce some obscure terminology that neuroscientists use to label the various actors in our tug-of-war drama (going from left to right in the Figure): • the inhibitory conductance (g is the symbol for a conductance, and i indicates inhibition) this is the total strength of the inhibitory input (i.e., how strong the inhibitory guy is tugging), and plays a major role in determining how strong of an inhibitory current there is. This corresponds biologically to the proportion ofCCNBook/Neuron 13 inhibitory ion channels that are currently open and allowing inhibitory ions to flow (these are chloride or Cl- ions in the case of GABA inhibition, and potassium or K+ ions in the case of leak currents). For electricity buffs, the conductance is the inverse of resistance most people find conductance more intuitive than resistance, so we'll stick with it. • the inhibitory driving potential in the tug-of-war metaphor, this just amounts to where the inhibitory guy happens to be standing relative to the electrical potential scale that operates within the neuron. Typically, this value is around -75mV where mV stands for millivolts one thousandth (1/1,000) of a volt. These are very small electrical potentials for very small neurons. • the action potential threshold this is the electrical potential at which the neuron will fire an action potential output to signal other neurons. This is typically around -50mV. This is also called the firing threshold or the spiking threshold, because neurons are described as "firing a spike" when they get over this threshold. • the membrane potential of the neuron (V = voltage or electrical potential, and m = membrane). This is the current electrical potential of the neuron relative to the extracellular space outside the neuron. It is called the membrane potential because it is the cell membrane (thin layer of fat basically) that separates the inside and outside of the neuron, and that is where the electrical potential really happens. An electrical potential or voltage is a relative comparison between the amount of electric charge in one location versus another. It is called a "potential" because when there is a difference, there is the potential to make stuff happen. For example, when there is a big potential difference between the charge in a cloud and that on the ground, it creates the potential for lightning. Just like water, differences in charge always flow "downhill" to try to balance things out. So if you have a lot of charge (water) in one location, it will flow until everything is all level. The cell membrane is effectively a dam against this flow, enabling the charge inside the cell to be different from that outside the cell. The ion channels in this context are like little tunnels in the dam wall that allow things to flow in a controlled manner. And when things flow, the membrane potential changes In the tug-of-war metaphor, think of the membrane potential as the flag attached to the rope that marks where the balance of tugging is at the current moment. • the excitatory driving potential this is where the excitatory guy is standing in the electrical potential space (typically around 0 mV). • the excitatory conductance this is the total strength of the excitatory input, reflecting the proportion of excitatory ion channels that are open (these channels pass sodium or Na+ ions our deepest thoughts are all just salt water moving around).CCNBook/Neuron 14 Figure 2.4 shows specific cases in the tug-of-war scenario. In the first case, the excitatory conductance is very low (indicated by the small size of the excitatory guy), which represents a neuron at rest, not receiving many excitatory input signals from other neurons. In this case, the inhibition/leak pulls much more strongly, and keeps the membrane potential (Vm) down near the -70mV territory, which is also called the resting potential of the neuron. As such, it is below the action potential threshold , and so the neuron does not output any signals itself. Everyone is just chillin'. In the next case (Figure 2.4b), the Figure 2.4: Specific cases in the tug-of-war scenario. excitation is as strong as the inhibition, and this means that it can pull the membrane potential up to about the middle of the range. Because the firing threshold is toward the lower-end of the range, this is enough to get over threshold and fire a spike The neuron will now communicate its signal to other neurons, and contribute to the overall flow of information in the brain's network. The last case (Figure 2.4c) is particularly interesting, because it illustrates that the integration process is fundamentally relative what matters is how strong excitation is relative to the inhibition. If both are overall weaker, then neurons can still get over firing threshold. Can you think of any real-world example where this might be important? Consider the neurons in your visual system, which can experience huge variation in the overall amount of light coming into them depending on what you're looking at (e.g., compare snowboarding on a bright sunny day versus walking through thick woods after sunset). It turns out that the total amount of light coming into the visual system drives both a "background" level of inhibition, in addition to the amount of excitation that visual neurons experience. Thus, when it's bright, neurons get greater amounts of both excitation and inhibition compared to when it is dark. This enables the neurons to remain in their sensitive range for detecting things despite large differences in overall input levels. Computing Activation Output The membrane potential Vm is not communicated directly to other neurons instead it is subjected to a threshold and only the strongest levels of excitation are then communicated, resulting in a much more efficient and compact encoding of information in the brain. In human terms, neurons are sensitive to "TMI" (too much information) constraints, also known as "Gricean Maxims" wikipedia link 1 e.g., only communicate relevant, important information. Actual neurons in the Neocortex compute discrete spikes or action potentials, which are very brief ( 1 ms) and trigger the release of neurotransmitter that then drives the excitation or inhibition of the neurons they are sending to. After the spike, the membrane potential Vm is reset back to a low value (at or even below the resting potential), and it must then climb back up again to the level of the threshold before another spike can occur. This process results in different rates of spiking associated with different levels of excitation it is clear from eletrophysiologicalCCNBook/Neuron 15 recordings of neurons all over the neocortex that this spike rate information is highly informative about behaviorally and cognitively relevant information. There remains considerable debate about the degree to which more precise differences in spike timing contain additional useful information. In our computer models, we can simulate discrete spiking behavior directly in a very straightforward way (see below for details). However, we often use a rate code approximation instead, where the activation output of the neuron is a real valued number between 0-1 that corresponds to the overall rate of neural spiking. We typically think of this rate code as reflecting the net output of a small population of roughly 100 neurons that all respond to similar information the neocortex is organized anatomically with microcolumns of roughly this number of neurons, where all of the neurons do indeed code for similar information. Use of this rate code activation enables smaller-scale models that converge on a stable interpretation of the input patterns rapidly, with an overall savings in computational time and model complexity. Nevertheless, there are tradeoffs in using these approximations, which we will discuss more in the Networks and other chapters. Getting the rate code to produce a good approximation to discrete spiking behavior has been somewhat challenging in the Leabra framework, and only recently has a truly satisfactory solution been developed, which is now the standard in the emergent software. Mathematical Formulations Now you've got an intuitive understanding of how the neuron integrates excitation and inhibition. We can capture this dynamic in a set of mathematical equations that can be used to simulate neurons on the computer. The first set of equations focuses on the effects of inputs to a neuron. The second set focuses on generating outputs from the neuron. We will cover a fair amount of mathematical ground here. Don't worry if you don't follow all of the details. As long as you follow conceptually what the equations are doing, you should be able to build on this understanding when you get your hands on the actual equations themselves and explore how they behave with different inputs and parameters. You will see that despite all the math, the neuron's behavior is indeed simple: the amount of excitatory input determines how excited it gets, in balance with the amount of inhibition and leak. And the resulting output signals behave pretty much as you would expect. Computing Inputs We begin by formalizing the "strength" by which each side of the tug-of-war pulls, and then show how that causes the Vm "flag" to move as a result. This provides explicit equations for the tug-of-war dynamic integration process. Then, we show how to actually compute the conductance factors in this tug-of-war equation as a function of the inputs coming into the neuron, and the synaptic weights (focusing on the excitatory inputs for now). Finally, we provide a summary equation for the tug-of-war which can tell you where the flag will end up in the end, to complement the dynamical equations which show you how it moves over time. Neural Integration The key idea behind these equations is that each guy in the tug-of-war pulls with a strength that is proportional to both its overall strength (conductance), and how far the "flag" (Vm) is away from its position (indicated by the driving potential E). Imagine that the tuggers are planted in their position, and their arms are fully contracted when the Vm flag gets to their position (E), and they can't re-grip the rope, such that they can't pull any more at this point. To put this idea into an equation, we can write the "force" or current that the excitatory guy exerts as: • • excitatory current: The excitatory current is (I is the traditional term for an electrical current, and e again for excitation), and it is the product of the conductance times how far the membrane potential is away from the excitatory driving potential. If then the excitatory guy has "won" the tug of war, and it no longer pulls anymore, and the current goes toCCNBook/Neuron 16 zero (regardless of how big the conductance might be anything times 0 is 0). Interestingly, this also means that the excitatory guy pulls the strongest when the Vm "flag" is furthest away from it i.e., when the neuron is at its resting potential. Thus, it is easiest to excite a neuron when it's well rested. The same basic equation can be written for the inhibition guy, and also separately for the leak guy (which we can now reintroduce as a basic clone of the inhibition term): • • inhibitory current: • • leak current: (only the subscripts are different). Next, we can add together these three different currents to get the net current, which represents the net flow of charged ions across the neuron's membrane (through the ion channels): • • net current: So what good is a net current? Recall that electricity is like water, and it flows to even itself out. When water flows from a place where there is a lot of water to a place where there is less, the result is that there is less water in the first place and more in the second. The same thing happens with our currents: the flow of current changes the membrane potential (height of the water) inside the neuron: • • update of membrane potential due to net current: ( is the current value of Vm, which is updated from value on the previous time step , and the is a rate constant that determines how fast the membrane potential changes it mainly reflects the capacitance of the neuron's membrane). The above two equations are the essence of what we need to be able to simulate a neuron on a computer It tells us how the membrane potential changes as a function of the inhibitory, leak and excitatory inputs given specific numbers for these input conductances, and a starting Vm value, we can then iteratively compute the new Vm value according to the above equations, and this will accurately reflect how a real neuron would respond to similar such inputs To summarize, here's a single version of the above equations that does everything: • For those of you who noticed the issue with the minus sign above, or are curious how all of this relates to Ohm's law and the process of diffusion, please see Electrophysiology of the Neuron. If you're happy enough with where we've come, feel free to move along to finding out how we compute these input conductances, and what we then do with the Vm value to drive the output signal of the neuron.CCNBook/Neuron 17 Computing Input Conductances The excitatory and inhibitory input conductances represent the total number of ion channels of each type that are currently open and thus allowing ions to flow. In real neurons, these conductances are typically measured in nanosiemens (nS), which is siemens (a very small number neurons are very tiny). Typically, neuroscientists divide these conductances into two components: • ("g-bar") a constant value that determines the maximum conductance that would occur if every ion channel were to be open. • a dynamically changing variable that indicates at the present moment, what fraction of the total number of ion channels are currently open (goes between 0 and 1). Thus, the total conductances of interest are written as: • • excitatory conductance: • • inhibitory conductance: • • leak conductance: (note that because leak is a constant, it does not have a dynamically changing value, only the constant g-bar value). This separation of terms makes it easier to compute the conductance, because all we need to focus on is computing the proportion or fraction of open ion channels of each type. This can be done by computing the average number of ion channels open at each synaptic input to the neuron: • where is the activity of a particular sending neuron indexed by the subscript i, is the synaptic weight strength that connects sending neuron i to the receiving neuron, and n is the total number of channels of that type (in this case, excitatory) across all synaptic inputs to the cell. As noted above, the synaptic weight determines what patterns the receiving neuron is sensitive to, and is what adapts with learning this equation shows how it enters mathematically into computing the total amount of excitatory conductance. The above equation suggests that the neuron performs a very simple function to determine how much input it is getting: it just adds it all up from all of its different sources (and takes the average to compute a proportion instead of a sum so that this proportion is then multiplied by g_bar_e to get an actual conductance value). Each input source contributes in proportion to how active the sender is, multiplied by how much the receiving neuron cares about that information the synaptic weight value. We also refer to this average total input as the net input. The same equation holds for inhibitory input conductances, which are computed in terms of the activations of inhibitory sending neurons, times the inhibitory weight values. There are some further complexities about how we integrate inputs from different categories of input sources (i.e., projections from different source brain areas into a given receiving neuron), which we deal with in the optional subsection: Net Input Detail. But overall, this aspect of the computation is relatively simple and we can now move on to the next step, of comparing the membrane potential to the threshold and generating some output.